In this paper, we have derived a global maximum principle for progressive optimal control of partially observed forward-backward stochastic systems with random jumps, with a hierarchical method. Compared to [1]}, we extend it to the case with partial information and random jumps. Compared to [2]}, we extend it to the partially observed forward-backward stochastic system. Compared to [3]}, we extend it to the forward-backward system with random jumps. Compared to [4]}, we extend it to the forward-backward system with both Brownian motion and Poisson martingale measure. We should notice that, different from [3]} and [4]}, we extend the continuous observation equation into discontinuous case, in which Brownian motion and Poisson random measure are both considered. Meanwhile, the existence and uniqueness result of the solution to fully coupled FBSDEPs in \(L^\beta \) -space and its \(L^\beta (\beta \ge 2)\) -estimate are given to prepare for the maximum principle, and the non-linear filtering equation for partially observed stochastic system is derived for obtaining the optimal control's state estimate feedback for the LQ case. Also, different from the work in [7]}, we extend it to the general non-linear forward-backward system and observation equation with Poisson random measure, where the control domain is non-convex, instead of the LQ stochastic system with Poisson process. Moreover, we have given an explicit form of the optimal control by introducing some ODEs.
